Why is Warp 1 referred to as the speed of light?
Since Warp drive is an FTL drive to begin with and FTL signifies 'Faster than Light' ... not 'Exactly as light'.
If you want to be
that accurate about it, let's remember nobody has ever said warp would be "faster than light". (Nobody except you and me and a couple of thousand other fans, and other people who are not really part of the Star Trek universe, that is.) Starfleet might very well have some fancier term for it - because, at least per ST:TMP, warp drive also allows travel below the speed of light.
To be sure, though, nobody on screen has quite claimed that warp 1 would be exactly lightspeed, either. It might be quite a bit higher than that, really, considering that in ENT, Boomer ships incapable of warp 2 are still capable of interstellar flight in practical timeframes, less than the decades or centuries that lightspeed travel would suggest.
From what I gathered watching that scene ... they had to have been very close to the speed of light
It certainly didn't seem or sound like it. No optical effects associated with near-lightspeed, no time dilation, nothing of the sort. (If you go to Trekcore.com, there's an older version of the ST:FC script that's quite different from the filmed one in certain details, one of them being Cochrane's flight - they have some relativistic stuff there. But not in the final version.)
We don't know if there exists a "critical velocity" for other, more modern types of warp engine. Certainly there are warp drives that don't require sublight acceleration, as we have seen the ships in TNG, DS9, VOY and even the TOS movies jump to warp from essentially standstill. See for example ST5:TFF, and the bit where Chekov goes to warp just in the nick of time to avoid a Klingon torpedo.
Timo Saloniemi